What is the difference between a coordinate geometry proof and a proof method that does not use coordinates? When would it be ok to use a coordinate proof instead of another method?

A coordinate proof uses a coordinate system. This type of proof allows you to use algebraic techniques instead of only synthetic techniques. You can use coordinate proofs almost anytime that you could do a synthetic proof. Sometimes the algebra is "messy", and sometimes the algebra is too hard to do. Using coordinate geometry allows you to use equations of lines, the midpoint formula, the distance formula, equations for circles, ellipses, etc...

For example, you can prove the Pythagorean theorem using a coordinate proof: without loss of generality place the right angle of the triangle at the origin with the legs lying on the positive x and y axes. Let the vertex A be at (a,0), the vertex B be at (0,b) and the vertex C at (0,0) with `bar(AC)_|_bar(BC)` . Then by the distance formula